In this setting, \(n = m\)  and \(\mu _{ij}\)  denotes the mutual preference value between \(i\)  and \(j\)  (e.g., the probability of a “match” between \(i\)  and \(j\) ). Following [1]}, we extend our previous framework to reciprocal recommendation by introducing the two-sided utility of a user \(i\) , which sums the utility \(\overline{u}_i(P)\)  derived by \(i\)  from the recommendations it gets, and the utility \(\overline{v}_i(P)\)  from being recommended to other users:
\(&u_i(P) = \overline{u}_i(P) + \overline{v}_i(P) \, = \sum _{i,j} (\mu _{ij}+ \mu _{ji}) P_{ij}^\intercal {b}\\&\text{where } ~~\overline{u}_i(P) = \sum _{i=1}^n \mu _{ij}P_{ij}^\intercal {b}\quad \text{ and } \quad \overline{v}_i(P) = \sum _{j=1}^n \mu _{ij}P_{ij}^\intercal {b}\,.\) 
